A survey asks teachers and students whether they would like the new school mascot to be a Viking or a Patriot. This table shows the results.
\begin{tabular}{|l|c|c|c|}
\hline & Vikings & Patriots & Total \\
\hline Students & 80 & 20 & 100 \\
\hline Teachers & 5 & 20 & 25 \\
\hline Total & 85 & 40 & 125 \\
\hline
\end{tabular}
A person is randomly selected from those surveyed.
Are being a student and preferring "Viking" independent events? Why or why not?
A. Yes, they are independent because [tex]P(\text{student}) = 0.80[/tex] and [tex]P(\text{student} \mid \text{Viking}) = 0.68[/tex].
B. No, they are not independent because [tex]P(\text{student}) = 0.80[/tex] and [tex]P(\text{student} \mid \text{Viking}) = 0.94[/tex].
C. Yes, they are independent because [tex]P(\text{student}) = 0.80[/tex] and [tex]P(\text{student} \mid \text{Viking}) = 0.94[/tex].
D. No, they are not independent because [tex]P(\text{student}) = 0.80[/tex] and [tex]P(\text{student} \mid \text{Viking}) = 0.68[/tex].
Answer (2)
Calculate the probability of being a student: P ( St u d e n t ) = 125 100 = 0.80 .
Calculate the conditional probability of being a student given a preference for Viking: P ( St u d e n t ∣ Vikin g ) = 85 80 ≈ 0.94 .
Compare the two probabilities: P ( St u d e n t ) = P ( St u d e n t ∣ Vikin g ) .
Conclude that the events are not independent: N o .
Explanation
Understand the problem We are given a table that summarizes the results of a survey asking teachers and students whether they would like the new school mascot to be a Viking or a Patriot. We want to determine if the events 'being a student' and 'preferring Viking' are independent.
Define independence Two events are independent if and only if P ( A ∣ B ) = P ( A ) . In this case, we need to check if P ( St u d e n t ∣ Vikin g ) = P ( St u d e n t ) .
Calculate P(Student) First, let's calculate P ( St u d e n t ) . The total number of people surveyed is 125, and the number of students is 100. Therefore, P ( St u d e n t ) = 125 100 = 0.80
Calculate P(Student | Viking) Next, let's calculate P ( St u d e n t ∣ Vikin g ) . This is the probability that a person is a student given that they prefer Viking. The total number of people who prefer Viking is 85, and the number of students who prefer Viking is 80. Therefore, P ( St u d e n t ∣ Vikin g ) = 85 80 ≈ 0.94
Compare the probabilities Now, we compare P ( St u d e n t ) and P ( St u d e n t ∣ Vikin g ) . We have P ( St u d e n t ) = 0.80 and P ( St u d e n t ∣ Vikin g ) ≈ 0.94 . Since 0.80 = 0.94 , the events 'being a student' and 'preferring Viking' are not independent.
Choose the correct answer Since the events are not independent, we look for an answer choice that says they are not independent and provides the correct probabilities. Option B states that the events are not independent because P ( St u d e n t ) = 0.80 and P ( St u d e n t ∣ Vikin g ) = 0.94 . This matches our calculations and conclusion.
Examples
Understanding independence is crucial in many real-world scenarios. For example, in marketing, businesses want to know if a customer's purchase of one product is independent of their purchase of another. If they are not independent, the business can use this information to create targeted marketing campaigns. Similarly, in medical research, understanding whether a risk factor (like smoking) is independent of a disease (like lung cancer) helps determine causal relationships and inform public health policies. In this case, we determined that being a student and preferring Vikings are not independent events, which might suggest that students have different preferences than teachers regarding the school mascot.
The events 'being a student' and 'preferring Vikings' are not independent since the probability of being a student given a preference for Vikings differs from the overall probability of being a student. Thus, the correct choice is B: they are not independent because P(student) = 0.80 and P(student | Viking) ≈ 0.94.
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